"Frontmatter". In: Analysis of Financial Time Series

434 MCMC METHODSThe probabilityP(s j | .) ∝n∏f (a t | H)P(s j | S − j ).t= jP(s j = i | S − j ) = P(s j = i | s j−1 , s j+1 ), i = 1, 2can be computed by the Markov transition probabilities in Eq. (10.41). In addition,assuming s j = i, one can compute h t for t ≥ j recursively. The relevantlikelihood function, denoted by L(s j ),isgivenby[ ]n∏n∑L(s j = i) ≡ f (a t | H) ∝ exp( f ji ), f ji = − 1 ln(h t ) + a2 t,2 h tt= jt= jfor i = 1 and 2, where a t = r t − β 1√ht if s t = 1anda t = r t − β 2√htotherwise. Consequently, the conditional posterior probability of s j = 1isP(s j = 1 | .)=P(s j = 1 | s j−1 , s j+1 )L(s j = 1)P(s j = 1 | s j−1 , s j+1 )L(s j = 1) + P(s j = 2 | s j−1 , s j+1 )L(s j = 2) .The state s j can then be drawn easily using a uniform distribution on the unitinterval [0, 1].Remark: Since s j and s j+1 are highly correlated when e 1 and e 2 are small, itis more efficient to draw several s j jointly. However, the computation involved inenumerating the possible state configurations increases quickly with the number ofstates drawn jointly.Example 10.5. In this example, we consider the monthly log stock returns ofGeneral Electric Company from January 1926 to December 1999 for 888 observations.The returns are in percentages and shown in Figure 10.11(a). For comparisonpurpose, we start with a GARCH-M model for the series and obtainr t = 0.182 √ h t + a t , a t = √ h t ɛ t ,h t = 0.546 + 1.740h t−1 − 0.775h t−2 + 0.025at−1 2 , (10.42)where r t is the monthly log return and {ɛ t } is a sequence of independent Gaussianwhite noises with mean zero and variance 1. All parameter estimates are highly significantwith p values less than 0.0006. The Ljung–Box statistics of the standardizedresiduals and their squared series fail to suggest any model inadequacy. It is reassuringto see that the risk premium is positive and significant. The GARCH model in